In the diagrams only 2 chromosomes are shown for simplicity.
The pink/red represents the maternal or queen's component and the blue the paternal or drone's component.

Crossing over is the process by which the two chromosomes of a homologous pair exchange equal segments with each other.
Crossing over occurs in Prophase I during meiosis I. At that stage each chromosome has replicated into two strands - the sister chromatids.
The two homologous chromosomes of a pair synapse, or come together. While the chromosomes are synapsed, breaks occur at corresponding points in two of the non-sister chromatids, i.e. in one chromatid of each chromosome. Since the chromosomes are homologous, breaks at corresponding points mean that the segments that are broken off contain corresponding genes, i.e. alleles.
The broken sections are then exchanged between the chromosomes to form complete new units, and each new recombined chromosome of the pair can go to a different daughter cell which may then be contained in the egg.
Crossing over results in recombination of genes found on the same chromosome, called linked genes, that would otherwise always be transmitted together. Because the frequency of crossing over between any two linked genes is proportional to the chromosomal distance between them, crossing over frequencies are used to construct genetic, or linkage, maps of genes on chromosomes.

Recombination
As a result of genes crossing over a new combination of alleles is produced on the chromosome - this is called recombination.
The rate of recombination in the honeybee is the highest found in the higher animals.

The process by which the chromosome number is halved during gamete formation is called meiosis. In meiosis in the honey bee, a cell containing the diploid number of chromosomes (32) is converted into four cells, each having the haploid number of chromosomes (16). As a result of what happens in meiosis each of these cells is genetically distinct i.e. each honey bee, whether it be queen, worker, or drone, is genetically unique. Only one of these haploid cells appears in the egg the other three are reabsorbed.
Two major phases of meiosis occur: meiosis I and meiosis II. During meiosis I, a single cell divides into two. During meiosis II, those two cells each divide again.

Visual and narrated animation of Meiosis
Each phase of meiosis which can be summarized as follows:
before meiosis begins in the preceding interphase, DNA replication takes place.
Meiosis I:
a honeybee cell contains 32 chromosomes, or 64 chromatids. Meiosis I proceeds through the following phases:Prophase I: The chromatids shorten and thicken and become visible under a microscope.
During prophase I, the two homologous chromosomes associate. Because each homologous chromosome consists of two chromatids, there are four chromatids aligned next to one another. This combination of four chromatids forms a tetrad; the coming together is synapsis.
After synapsis has taken place, the process of crossing over occurs. In this process, segments of DNA from one chromatid in the tetrad pass to another chromatid in the tetrad. They result in a genetically new chromatid. After crossing over has taken place, it is important to realize that the four chromatids of the tetrad are genetically different from the original four chromatids.Metaphase I: the tetrads align randomly on the equatorial plane. The centromeres attach to spindle fibers, which extend from the poles of the cell. (There are two possibilities; only one is shown on the diagram).Anaphase I: the homologous chromosomes separate. One homologous chromosome (consisting of two chromatids) moves to one pole, while the other homologous chromosome (consisting of two chromatids) moves to the other pole. The result is that 16 chromosomes (each consisting of two chromatids) move to one pole, and 16 chromosomes (each consisting of two chromatids) move to the other pole. Essentially, the chromosome number of the cell is halved. Telophase I: the nucleus reorganizes, the chromosomes become chromatin, and a cytoplasmic division into two cells takes place. Each daughter cell (with 16 chromosomes each consisting of two chromatids) then enters interphase, during which there is no duplication of the DNA.
Meiosis II
In meiosis II, the two cells, formed in Meiosis I, containing 32 chromatids undergo division into four cells, each with 16 chromosomes. Meiosis II proceeds through the following phases:Prophase II: The chromatin material condenses, and each chromosome contains two chromatids attached by the centromere. The 16 chromatid pairs, a total of 32 chromatids, then move to the equatorial plane.Metaphase II: the 16 chromatid pairs gather at the center of the cell prior to separation. Anaphase II: the centromeres divide, and the 32 chromatids become 32 chromosomes. Then the 32 chromosomes separate from one another. Spindle fibers move one chromosome from each pair to one pole of the cell and the other member of the pair to the other pole. In all, 16 chromosomes move to each pole. Telophase II: the chromosomes gather at the poles of the cells and become indistinct. Again, they form a mass of chromatin. The nuclear envelope develops, the nucleoli reappear, and the cells undergo cytokinesis (cell division). As a result 4 cells are formed each containing just one set of chromosomes i.e. they are haploid.

Only one of these haploid cells then is found in the egg; the other three are reabsorbed by the system.